Hello and Welcome to justanswer...
From what you have described this sounds like an Ect sensor problem... This is the engine coolant sensor - not the one the controls the gauge - but the one that sends signals to the cars Ecu.. main computer - re. the temperature of the engine.
The Ecu then sends commands to the fueling and ignition systems according. If the sensor is faulty and thus sends incorrect , or no info to the ecu as above, the fuel and air mixture will be incorrect and thus do as you describe resulting in poor running and no start situations and / or the symptoms you describe..
If you remember the old auto and manual choke systems...it works like that , only electronically. If and when the choke was in the wrong position for the temp of the engine, the engine either wouldn't start or coughed and spluttered... exactly the same.
So, without the aid of plug in diagnostics at the moment and the results thereof - its well worth renewing that sensor. Its usually located on the inlet thermostat housing , inlet manifold or thereabouts depending on model and has a two wires pin connector going to it.
Always buy one from a dealer / genuine unit. If you are unsure as to its position - ask the parts dept; when you are in there, to show you on the screen where it is located ....
Feedback and any Bonuses are always appreciated too. Thank you.